In a message dated 97-08-11 16:24:26 EDT, Jerry Rosen writes, quoting Mark Snyder and me,
> >I really like all these suggestions for doing the limit of radical x - over > >x, but I guess the real question is WHY are you doing the limit? > > Well, *I* did it because I was asked to... > > >Possible answers: > > > >(1) You need to know the answer to do some other problem. In that case graph > >it on your calculator and get on with the other problem. > > >(2)You are a good teacher and want to show your class how to do limits (a) > >using algebraic manipulation, (b) using a Taylor series, (c) using a table of > >values (d) using Mathematica or (e) etc. In this case use (a) algebraic > >manipulation, (b) a Taylor series, (c) a table of values (d) Mathematica or > >(e) etc.
<< > (1) doesn't actually work. Graphing something on your calculator only > gives you an approximate answer, or a hint as to what the real answer is. > The same thing holds for numerical investigations of limits. Just because > the graph "looks like" it hits the number 1, or the number "seems to be > settling down to 1" doesn't mean that it does. It might hit the number > 1.00001, or 1 - (1/pi^89), and depending on the context that gave rise to > the problem, that difference might be significant. Or maybe it settles > down to 1 until you get to within 10^(-50) of the value, and then it jumps > up to pi^2/6. >>
I know all that. I know you can find examples where a calculator fails. BUT then you go on to talk of engineers and exact answers in the same paragraph. Is it important to an engineer in a practical situation if the limit is 1 or 1 - (1/pi^89) ??? There are of course calculator and CAS on computers (same thing) that find the "exact" value.
<< Graphing (or numerical analysis) is of little, if any, help when you have a > function that depends on two (or more) variables, and you want the limit as > one of the variables approaches, say, infinity. This is *extremely* common > in physics problems. In that case, you are interested in what the limiting > *function* is, and algebraic tricks, LHop, and Taylor series are the only > approaches that give you that. >>
Of course if you can't find this particular limit on a calculator (or CAS) then you can't find it on a calculator (or CAS) and you'll have to find it some other way. BTW the TI92 can find limits as x approaches infinity, as can may computer programs.
And I think that there is a (4)
> > (4) You are a good teacher who has misplaced the phone numbers of the > people who your students will be working with ten years from now, so you > can't call them up to ask which approach each of your students will need, > so you cover all the approaches (even to the point of having your students > think you are just "showing off"). This way they will be prepared for > that, as well as for the AB test. > >>
Your number 4 is my number 2. Certainly a good teacher will show all the methods mentioned. That was the point of my original question " WHY are you doing the limit?"